| Description |
1 online resource (xi, 227 pages) : illustrations |
| Series |
Undergraduate texts in mathematics |
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Undergraduate texts in mathematics.
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| Contents |
Preface -- 1. Straightedge and compass -- 1.1. Euclid's construction axioms -- 1.2. Euclid's construction of the equilateral triangle -- 1.3. Some basic constructions -- 1.4. Multiplication and division-- 1.5. Similar triangles -- 1.6. Discussion -- 2. Euclid's approach to geometry -- 2.1. The parallel axiom -- 2.2. Congruence axioms -- 2.3. Area and equality -- 2.4. Area of parallelograms and triangles -- 2.5. The Pythagorean theorem -- 2.6. Proof of the Thales theorem -- 2.7. Angles in a circle -- 2.8. The Pythagorean theorem revisited -- 2.9. Discussion -- 3. Coordinates -- 3.1. The number line and the number plane -- 3.2. Lines and their equations -- 3.3. Distance -- 3.4. Intersections of lines and circles -- 3.5. Angle and slope -- 3.6. Isometries -- 3.7. The three reflections theorem -- 3.8. Discussion -- 4. Vectors and euclidean spaces -- 4.1. Vectors -- 4.2. Direction and linear independence -- 4.3. Midpoints and centroids -- 4.4. The inner product -- 4.5. Inner product and cosine -- 4.6. The triangle inequality -- 4.7. Rotations, matrices, and complex numbers -- 4.8. Discussion |
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5. Perspective -- 6.1. Perspective drawing -- 5.2. Drawing with straightedge alone -- 5.3. Projective plane axioms and their models -- 5.4. Homogeneous coordinates -- 5.5. Projection -- 5.6. Linear fractional functions -- 5.7. The cross-ratio -- 5.8. What is special about the cross-ratio? -- 5.9. Discussion -- 6. Projective planes -- 6.1. Pappus and Desargues revisited -- 6.2. Coincidences -- 6.3. Variations on the Desargues theorem -- 6.4. Projective arithmetic -- 6.5. The field axioms -- 6.6. The associative laws -- 6.7. The distributive law -- 6.8. Discussion -- 7. Transformations -- 7.1. The group of isometries of the plane -- 7.2. Vector transformations -- 7.3. Transformations of the projective line -- 7.4. Spherical geometry -- 7.5. The rotation group of the sphere -- 7.6. Representing space rotations by quaternions -- 7.7. A finite group of space rotations -- 7.8. The group S³ and RP³ -- 7.9. Discussion -- 8. Non-Euclidean geometry -- 8.1. Extending the projective line to a plane -- 8.2. Complex conjugation -- 8.3. Reflections and Möbius transformations -- 8.4. Preserving non-Euclidean lines -- 8.5. Preserving angle -- 8.6. Non-Euclidean distance -- 8.7. Non-Euclidean translations and rotations -- 8.8. Three reflections or two involutions -- 8.9. Discussion -- References -- Index |
| Summary |
"The Four Pillars of Geometry approaches geometry in four different ways, devoting two chapters to each. This makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic. Not only does each approach offer a different view; the combination of viewpoints yields insights not available in most books at this level. For example, it is shown how algebra emerges from projective geometry, and how the hyperbolic plane emerges from the real projective line." "All readers are sure to find something new in this attractive text, which is abundantly supplemented with figures and exercises. This book will be useful for an undergraduate geometry course, a capstone course, or a course aimed at future high school teachers."--Jacket |
| Bibliography |
Includes bibliographical references (pages 213-214) and index |
| Notes |
Print version record |
| In |
Springer e-books |
| Subject |
Geometry -- Textbooks
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Geometry.
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geometry.
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Matrix theory.
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Mathematics.
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Geometry.
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Linear and Multilinear Algebras, Matrix Theory.
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Geometría -- Libros de texto
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Geometry
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Meetkunde.
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| Genre/Form |
Textbooks
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Textbooks.
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| Form |
Electronic book
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| LC no. |
2005929630 2005923256 |
| ISBN |
9780387290522 |
|
0387290524 |
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0387255303 |
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9780387255309 |
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